Debug advice Kenmore coldspot 106-59422801 stopped refrigerating

Debugging advice requested.
I have never debugged a refrigerator in my life.
I never even looked at the back of one before.
2010 Sears Kenmore coldspot 106-59422801 refrigerator/freezer just stopped
working at the same time that a periodic "humming and then clicking" noise
started happening.
Inside temperature this morning was 59F degrees in the frig and 49F in the
freezer (outside temperature is about 70F).
Both dial settings are on cold (mid range) where they were always left.
There is no on/off switch that we know about.
Electricity is working (fan and lights are working).
Fan is blowing full time (dunno if it always did that).

http://i.cubeupload.com/dQOTKL.jpg
Coils are at room temperature and are dusty:

http://i.cubeupload.com/jgkVXv.jpg
Compressor may not be turning on (how can we tell)?

http://i.cubeupload.com/p2OiCZ.jpg
No reason to suspect icemaker water supply yet:

http://i.cubeupload.com/myOvgH.jpg
Top of compressor is hot to the touch but not so hot as to burn (but pretty
hot otherwise):

http://i.cubeupload.com/wAX37P.jpg
We hear a humming noise kick on every five minutes for about 10 or 20
seconds and then a click when it shuts off:
http://tinypic.com/r/e6abk7/9
The only things new are the humming noise, and the fact that the
refrigerator and freezer aren't working.
First aim is to figure out how this thing works.
I was expecting to see a "motor" for the black compressor "bulb" but there
is no motor I can see - but the top of the black compressor "bulb" is hot
to the touch - but the coils are not.
I think the solenoid at the compressor "bulb" is what is making the noise.
Any debugging advice?

http://i.cubeupload.com/aOLfge.jpg
I disconnected the spring (what is it there for?).
I disconnected the wiring.
I tried to pull the black cap outward.
I tried to pull the gray base outward.
Neither would budge.
Have you removed them before?
Is there a trick to removing them?

The spring is there to hold the cap in place. It is simply plugged into
the overload/relay body. See:
http://www.searspartsdirect.com/kenmore-refrigerator-parts/2264017/0046/106/model-10659422801/0582/0165000.html
It's not clear how the base is fastened, but you probably won't have to
remove it anyhow.
Bob

Sounds just exactly like a bad start relay
on the compressor. This is considered "sealed
system". If you have the owners manual, see
how long is the sealed system warranty. Probably
five years, so you're out of warranty.
If it's out of warranty, are you any good with
electrical repairs? Please write back, and we'll
continue the discussion.

Thanks for the advice.
The last thing I fixed with your help was the washing machine, who had an
F28/F11 which meant the motor control board was fried.
Before that, it was a non-operational furnace (bad thermistor), and before
that there were a host of things, most of which are working just fine now.
I have the frig unplugged to cool the black compressor "sealed system" down
(right now the black top of the "sealed system" is warm to the touch but
not hot).
A few questions of what I should expect, since I'm not sure that I
understand how the thing works yet.
Of course, I know that decompressing fluid freon (or whatever it uses) to a
gas makes the freon cold which makes the pipes that run through the
refrigerator cold, and then recompressing that recirculating gas back to a
liquid creates heat which is blown off by the fan - but I don't know the
mechanics of what I should expect (e.g., what pumps the fluid/gas through
the system? Gravity?).
So I have questions if I may ask...
0. Are there 3 things inside the "sealed unit"?
a. Motor
b. Compressor
c. Solenoid
1. Is the sealed unit a replaceable part?

http://i.cubeupload.com/wAX37P.jpg
(same pic as before)
2. Why would the top of the black sealed unit be hot to the touch?
3. Is there a motor overheat safety switch that may have tripped?
4. Are those black coils in the bottom supposed to be warm or cold?

http://i.cubeupload.com/jgkVXv.jpg
(same pic as before)
5. I don't see motion, but the vibrating can be felt on the solenoid on the
OUTSIDE of the sealed unit (it may be translating from inside though). What
is the purpose of that solenoid on the outside of the sealed unit?

http://i.cubeupload.com/p2OiCZ.jpg
(same pic as before)
In an hour or so, when the "sealed unit" has cooled down, I'll plug it back
in and report back.

CY: No, the motor and compressor are inside. The start relay (and some
have a relay and a capacitor) outside the sealed unit. The relay and cap
should never contact refrigerant. That said, the company probably
considers the relay and cap to be part of the sealed system.

CY: Yes. Though, it's often not cost effective. Requires refrigerant
pump, brazing, electrical, and more than that. Takes about three hours.

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